Combining On-Line Characterization Tools with Modern Software - MDPI

processes Article

Combining On-Line Characterization Tools with Modern Software Environments for Optimal Operation of Polymerization Processes Navid Ghadipasha 1 , Aryan Geraili 1 , Jose A. Romagnoli 1, *, Carlos A. Castor, Jr. 2 , Michael F. Drenski 3 and Wayne F. Reed 2 1 2 3

*

Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA 70803, USA; [email protected] (N.G.); [email protected] (A.G.) Department of Physics and Engineering Physics, Tulane University, New Orleans, LA 70118, USA; [email protected] (C.A.C.); [email protected] (W.F.R.) Advanced Polymer Monitoring Technologies, Inc., New Orleans, LA 70125, USA; [email protected] Correspondence: [email protected]; Tel.: +1-225-578-1377

Academic Editor: Masoud Soroush Received: 6 December 2015; Accepted: 5 February 2016; Published: 15 February 2016

Abstract: This paper discusses the initial steps towards the formulation and implementation of a generic and flexible model centric framework for integrated simulation, estimation, optimization and feedback control of polymerization processes. For the first time it combines the powerful capabilities of the automatic continuous on-line monitoring of polymerization system (ACOMP), with a modern simulation, estimation and optimization software environment towards an integrated scheme for the optimal operation of polymeric processes. An initial validation of the framework was performed for modelling and optimization using literature data, illustrating the flexibility of the method to apply under different systems and conditions. Subsequently, off-line capabilities of the system were fully tested experimentally for model validations, parameter estimation and process optimization using ACOMP data. Experimental results are provided for free radical solution polymerization of methyl methacrylate. Keywords: free radical polymerization; molar mass distribution; dynamic optimization; parameter estimation; online monitoring

1. Introduction In the polymer industry, batch and semi-batch reactors are widely used for the production of different classes of polymers. There is considerable economic incentive to develop real-time optimal operating policies that will result in the production of polymers with desired molecular properties. In the case of polymerization processes, the molar mass distribution (MMD) of a polymer is one of the most important quality control variables since many of the polymer end-use properties are directly dependent on the MMD. Some examples include the mechanical properties such as stiffness, strength and viscoelasticity [1–3]. Optimal operation in polymerization usually involves computing and accurately maintaining the optimal policies that can lead to a product with desired MMD and final conversion, while minimizing the total operation time. However, direct feedback control of MMD is difficult to achieve due to several reasons. The primary issue is the lack of on-line promising method for monitoring polymer properties. Monitoring requires continuous measurement of the reacting solution to make data available for closed-loop control or state estimation of the system. The ability to monitor polymerization reactions as they occur is necessary to ensure fulfillment of

Processes 2016, 4, 5; doi:10.3390/pr4010005

www.mdpi.com/journal/processes

Processes 2016, 4, 5

2 of 23

the proposed recipe. Most of the control strategies in the literature for polymeric systems are based on open-loop methods which heavily depend on the accuracy of the mathematical model and will produce unsatisfactory results in case of disturbances in the system. An extensive amount of work for feedback control of polymerization systems has been reported during last decades, most using an inferential technique to infer the polymer properties. Guyot et al. [4] used an on-line gas chromatographic analysis of the monomer mixture to control the copolymer composition in emulsion polymerization. Dimitratos et al. [5,6] developed a feedforward-feedback control strategy based on the use of an extended Kalman filter for state estimation. The Kalman filter method is based on a linear approximation of the nonlinear process [7] but has problems with stability and convergence [8–11]. For that reason many nonlinear methods have been developed. Hammori et al. [7] developed a nonlinear state observer that uses rate generation due to chemical reaction to obtain key parameters during free radical copolymerization. This technique is simpler to tune than Kalman filters. Kravaris et al. [12] utilized a general nonlinear feedforward-feedback control strategy based on temperature tracking to control copolymer composition. In order to control the nonlinear systems Model predictive control (MPC) [13–18] has been suggested. The main disadvantage of the MPC is that it cannot explicitly deal with plant model uncertainties. Common manipulated variables in polymerization processes include coolant or heating medium flow rates, gas or liquid flow rates for pressure control, feed rate of monomer, solvent, or initiator, and agitator speed [19]. Crowley et al. [20] introduced a new method for controlling the weight chain length distribution of polymer in batch free radical polymerization processes by manipulating temperature. Although temperature has a considerable effect on MMD, when the reaction is extremely exothermic, which is the case in many polymerization systems, changing temperature is not an efficient way to control MMD. In this case, MMD is controlled by manipulating the ratio of monomer to initiator or adding a chain transfer agent. This will give the extra degree of freedom to minimize the operation time and also conversion of monomer. The combined control of temperature and reagent flow has been studied by Ellis et al. [21]. They proposed a MMD estimator based on an extended Kalman filter to provide current estimates of the entire MMD based on measurements of monomer conversion obtained by on-line densitometry and periodic time delayed measurements of MMD from on-line size-exclusion chromatograph. Simultaneous change of temperature and monomer addition to control MMD was investigated and it was demonstrated that the proposed feedback control strategy is effective in rejecting disturbances. In general, controlling molar mass is best achieved by manipulating the monomer, initiator and chain transfer agent concentration [5,22,23]. More recently a platform for automatic continuous online monitoring of polymerization (ACOMP) was proposed [24]. ACOMP allows for automatic, continuous and model independent monitoring of polymerization reactions. It uses continuous dilution of a small stream from the reactor, in conjunction with light scattering, viscosity, ultraviolet absorption and refractometric detectors. This provides monomer and co-monomer conversion, weight average molar mass, weight average intrinsic viscosity, average composition drift and distribution, and certain measures of polydispersity [24,25]. The ability to monitor polymerization reactions as they occur brings several benefits. First, following kinetics and other reaction characteristics allows fundamental understanding of reaction mechanisms that can help accelerate the development of new polymer chemistries and processes. Second, monitoring the effects of changing reaction conditions, such as temperature, reagent types, and concentration, provides a concise means for bench scale or small pilot reaction optimization. This paper discusses the initial steps towards the formulation and implementation of a generic and flexible model-based framework for integrated simulation, estimation, optimization and feedback control of polymerization systems. The emphasis is on developing a comprehensive scheme which can be applied for the optimal operation of various polymeric systems and this was achieved by combining the powerful capabilities of the on-line monitoring system, ACOMP, with a modern simulation, estimation and optimization software environment. Using literature data, an initial validation of the framework for a number of cases of simulation and optimization is presented. This shows the


3 of 23

flexibility of the method to work under different systems and conditions. Subsequently, the off-line capabilities of the framework were fully tested experimentally for model validations, parameter estimation as well as for process optimization. Starting with kinetic information from literature a new set of kinetic parameters were obtained for our case study as well as confidence intervals for the estimated parameters. This illustrates the importance of this strategy to follow kinetics and other reaction characteristics allowing fundamental understanding of reaction mechanisms that can help in the development of new polymer chemistries and processes. An offline model-based optimization analysis was conducted and experimentally validated to determine the optimal temperature profile in conjunction with the optimal monomer and initiator flow rate in order to reach a final target polymer while minimizing the batch time. A brief analysis on the controllability of the system under feedback conditions was also performed through a combination of simulation and experimental work. 2. Model Centric Framework A shortcoming of using advanced operating and control strategies in polymerization processes is the unavailability of sensors able to determine online process status and provide corrective actions. The ACOMP platform will provide a leap towards integration of monitoring, control, and optimization tools in the control of complex industrial polymerization [26]. The proposed structure will forge initial links between ACOMP and advanced modelling and control principles and demonstrate unprecedented feedback control of polymerization reactions. The final goal is a self-contained intelligent system for advanced operation of polymerization processes where both ACOMP and the software environment exchange data seamless toward achieving target final products. The conceptual representation of the aforementioned framework for integrated simulation, estimation, optimization and feedback control of systems is illustrated in Figure 1. The modelling work was carried out using gPROMS modelling language (V4.1.0, Process Systems Enterprise Inc London, United Kingdom, 2015), providing a complete environment for modelling/analysis of complex systems. Among gPROMS other advantages are: (a) modelling and solution power; (b) multiple activities using the same model; (c) integrated steady state and dynamic capabilities; (d) parameter estimation potentials; (e) sophisticated optimization tools; and (f) structural model information for advanced process control implementations. The parameter estimation entity makes use of the data gathered from the experimental runs. It has the ability to estimate an unlimited number of parameters, using data from multiple dynamic experiments and ability to specify different variance models among the variables as well as among the different experiments. The optimization entity allows for the typical dynamic optimization problems arising from batch and/or semibatch operation to be formulated and implemented. One of the key issues is the connectivity of the software platform with the control system and ACOMP towards full integration. In this specific application gPROMS would need to be started from an external application and data will travel back and forth. An approach for this is to use the gSERVER API, transforming the application into the so-called gPROMS-based Application, or gBA or as an alternative via gO:RUN-xml. Both of these as well as open platform communication are currently explore in our experimental facilities.


4 of 23

Processes 2016, 4

4 of 24

Figure 1. Schematic of the integrated simulation, estimation, optimization and feedback control of Figure 1. Schematic of the integrated simulation, estimation, optimization and feedback control of polymerization systems. polymerization systems.

2.1. Process Modelling

2.1. Process Modelling

Modelling of polymerization processes is the other significant issue which plays a key role in the

Modelling of of polymerization processes is the other significant issue which plays a key role in development model-based control strategies. A number of well-known models have been introduced with attention control on the control of number and weight average molar mass [27,28]. the development of much model-based strategies. A number of well-known models have been Although weight average molaron mass usuallyof thenumber most important parameter in characterization of introduced with much attention theiscontrol and weight average molar mass [27,28]. the polymer chain length distribution, there are cases in which controlling average properties may Although weight average molar mass is usually the most important parameter in characterization be sufficient. For example, when a broad or bimodal distribution is desired [29]. A very attractive of thenot polymer chain length distribution, there are cases in which controlling average properties approach to characterize chain length distribution in batch free radical polymerization reactor has may not be sufficient. For example, when a broad or bimodal distribution is desired [29]. A very been presented by Crowley et al. [30,31]. This technique applies the method of finite molecular weight attractive approach to characterize chain length distribution in batch free radical polymerization reactor moments in conjunction with the kinetic rate equations to calculate the weight chain length has been presented et al. [30,31]. This technique applies of finite molecular distribution. Asby it Crowley was illustrated, it is feasible to control MMD the in amethod batch polymerization weight moments in conjunction with theinkinetic rateincorporates equations tothe calculate themoment weightto chain length process [20]. The modelling approach this work method of obtain distribution. As itdistribution was illustrated, it is feasible to control MMD inincludes a batch polymerization process chain length directly from the kinetic equations and the free volume theory to [20]. calculate the initiator in efficiency, termination and the propagation As a first trial, a chain detailed The modelling approach this work incorporates method rate. of moment to obtain length mechanistic model forthe solution methyl the methacrylate (MMA) in to batch and the distribution directly from kineticpolymerization equations andofincludes free volume theory calculate semi-batch reactors is developed and tested. This case selected since ample information initiator efficiency, termination and propagation rate. Asstudy a firstwas trial, a detailed mechanistic model for regarding the kinetic mechanisms and data for MMA polymerization was available in the literature solution polymerization of methyl methacrylate (MMA) in batch and semi-batch reactors is developed thus allowing faster prototyping. However, more complex systems will be considered in the future. and tested. This case study was selected since ample information regarding the kinetic mechanisms and data MMAMechanisms polymerization was available 2.1.1.for Reaction and Kinetic Equationsin the literature thus allowing faster prototyping. However, more complex systems will be considered in the future. The reaction mechanisms adopted consists of three important steps: Initiation, propagation and termination. Chain transfer monomer and solvent were also incorporated for better prediction of 2.1.1. Reaction Mechanisms andtoKinetic Equations the molar mass. The detailed mechanism is as follow: The reaction Initiationmechanisms adopted consists of three important steps: Initiation, propagation and

termination. Chain transfer to monomer and solvent 𝑘𝑘d were also incorporated for better prediction of (1) 𝐼𝐼 �� 2𝑅𝑅 the molar mass. The detailed mechanism is as follow: 𝑘𝑘i (2) Initiation 𝑅𝑅 + 𝑀𝑀 → 𝑃𝑃1 kd I Ñ 2R (1) Propagation k

Propagation

i R`M Ñ 𝑘𝑘p P1

𝑃𝑃n + 𝑀𝑀 �� 𝑃𝑃n+1 kp

Pn ` M Ñ Pn`1

(3)

(2)

(3)

Chain transfer to monomer and solvent k

fm Pn ` M Ñ Dn ` P1

(4)


5 of 23

k

fs Pn ` S Ñ Dn ` P1

Termination

(5)

k

tc Pn ` Pm Ñ Dn`m

(6)

k

td Pn ` Pm Ñ Dn ` Dm

(7)

where I denotes the initiator, R is the primary initiator radical, M is the monomer, S is the solvent and Pj and Dj are the corresponding growing live polymer radical and dead polymer. Under standard assumptions such as well-mixed reactor, quasi steady state assumptions (for the radicals) and long chain hypothesis, the following set of kinetic and dynamic equations describe the system: ` ˘ dNm “ ´ kp ` kfm P0 Nm ` Fm Cmf ´ Fout Cm dt

(8)

dNi “ ´kd Ni ` Fi Cif ´ Fout Ci dt

(9)

dNs “ ´kfs Ns P0 ` Fi Csif ` Fm Csmf ´ Fout Cs dt

(10)

1 d pλ0 Vq “ pkfm Nm ` ktd P0 V ` kfs Ns q αP0 ` ktc P02 V dt 2 ” ´ ¯ ı P d pλ1 Vq 0 “ pkfm Nm ` ktd P0 V ` kfs Ns q 2α ´ α2 ` ktc P0 N p1 ´ αq dt  „ ´ ¯ P0 d pλ2 Vq P0 “ pkfm Nm ` ktd P0 V ` kfs Ns q α3 ´ 3α2 ` 4α ` ktc P0 V pα ` 2q p1 ´ αq p1 ´ αq2 dt

(11) (12) (13)

where Nm = Cm V, Ni = Ci V, Ns = Cs V α“

kp Cm kp Cm ` kfm Cm ` kfs Cs ` ktc P0 ` ktd P0

(14)

d P0 “ «

λ wm V “ 1´ 1 ρp

ff´1 „

2 f Ci kd ktc ` ktd Nm wm Ns ws Nw ` ` i i ρm ρs ρi

(15)  (16)

Here Cm , Ci and Cs represent the concentrations of monomer, initiator and solvent in the reactor, respectively. V illustrates the volume of the content of the reactor, Fm and Fi are the volumetric flow rate of monomer and initiator respectively which are fed into the reactor in the semi batch mode. Fout is the constant flow rate out of the reactor for the ACOMP extraction stream. Cmf , Cif , Csif and Csmf are the concentration of monomer in the monomer feed stream, the initiator in the initiator feed stream and solvent in the initiator and monomer flow stream. P0 is the total concentration of live polymer which is obtained from the quasi steady state assumption. λ0 , λ1 and λ2 are the corresponding moments for the dead polymers, and α is the probability of propagation. f is the initiator efficiency and ρm , ρi , ρs and ρp are the densities of the monomer, initiator, solvent and polymer which are temperature dependent. kp , kd , kfm , kfs , ktc and ktd are the propagation, initiation, chain transfer to monomer, chain transfer to solvent, termination by combination and termination by disproportionation rate. Kinetic rate constants are all temperature dependent functions based on Arrhenius equation [28] and as we will see due to strong nonlinearity of the MMA system, the propagation and termination rate depend on conversion as well.


6 of 23

The conversion of the monomer is defined as the number of moles of monomer reacted in the tank divided by the total amount of monomer which has been loaded initially in the reactor and added by the semi-batch flow: şt şt Nm0 ` 0 Fm Cmf dt ´ Cm V ´ 0 Fout Cm dt X“ (17) şt Nm0 ` 0 Fm Cmf dt Here, Nm0 is the initial amount of monomer in the reactor. Number average and weight average molar mass of the polymers are calculated by considering only the moment of dead polymers and neglecting the live polymer concentration which is valid for low and medium conversions when the concentration of live polymer is negligible. Mn “ wm

λ1 λ2 , Mw “ wm λ0 λ1

(18)

2.1.2. Formalism for Gel, Glass and Cage Effects in MMA Polymerization In free radical polymerization, the mobility of the radicals decreases along the reaction due to increase in the viscosity of the reactor as more polymers are produced. This phenomenon which is called “gel effect” causes a reduction in the termination rate constant kt , and should be considered in the formulation of the model. At high conversion when even the motion of monomer is severely restricted the propagation rate kp , is also decreased. This glassy state in which the solution is highly viscous sets a limiting conversion on the polymerization process. In this work the correlation by [28,32] is used for both gel and glass effect. This can be written as: # gt “

0.10575 exp p17.15vf ´ 0.01715 pT ´ 273.15qq , vf ą vftc 2.3 ˆ 10´6 exp p75vf q , vf ď vftc # gp “

1, vf ą vfpc 7.1 ˆ 10´5 exp p171.53vf q , vf ď vfpc

(19)

(20)

Here, vftc and vfpc are the critical free volumes which are calculated as below: vftc “ 0.1856 ´ 2.965 ˆ 10´4 pT ´ 273.15q

(21)

vfpc “ 0.05

(22)

vf is the total free volume which is given by: vf “ φm vfm ` φs vfs ` φp vfp

(23)

` ˘ vfi “ 0.025 ` αi T ´ Tgi

(24)

where ϕi and Tgi are the volume fraction and the glass transition temperature of the polymer, solvent and monomer and αi is a constant. The values of the parameter for this case are shown in Table 1 [28]. For butyl acetate the corresponding glass transition temperature was considered as an adjustable parameter which has to be determined by parameter estimation. Table 1. Parameters for Methyl methacrylate gel and glass affect correlations. Parameter

MMA

Poly Methyl Methacrylate

Butyl Acetate

α Tg (K)

0.001 167

0.00048 387

0.001 #


7 of 23

The initiator efficiency f, appearing in Equation (15) describes the fraction of initiator free radicals which can successfully initiate the polymerization. Not all primary radicals can produce propagating chains. They execute many oscillations in “cages” before they diffuse apart and initiate a reaction. During the oscillations the radicals may also form an inactive species. Therefore, to account for the two competing phenomena, initiator efficiency is appended in the mathematical model. Initiator efficiency factor also decreases as the viscosity of the reactor solution rises. The free volume theory is used to model this relationship: ˆ ˆ ˙˙ 1 1 f “ f 0 exp ´C ´ (25) Vf Vfcr where f 0 is the initial initiator efficiency and C is a constant with the values of 0.53 and 0.006 for Azobisisobutyronitrile Fan et al. [33]. 2.1.3. Molar Mass Distribution In order to obtain complete representation of molecular weight distribution, a similar methodology proposed by Crowley et al. [30] based on finite weight fractions is applied with modifications for dealing with semi-batch operations. This method consists of dividing the entire polymer population into discrete intervals and calculating the weight fraction of polymer in each of the discrete intervals. By ignoring the concentration of live polymer, given that it is negligible compared with the dead polymer concentration, for each interval of (m,n) the polymer weight fraction is calculated with the following equations: řn řn iDi V i“m iDi V f pm, nq “ ř8 Or f pm, nq “ i“m (26) λ1 V i“2 iDi V The dynamic of weight fraction can then be obtained as: n d f pm, nq 1 ÿ d pDi Vq 1 d pλ1 Vq “ ´ f pm, nq i dt λ1 V dt λ1 V dt

(27)

i “m

where the right-hand side of Equation (27) represents the dynamic growth of dead polymers of length n which can be written according to the kinetic rate equation as below: ‰ d pDi Vq “ “ kfm Cm ` kfs Cs ` ktd Cp Pn V dt

(28)

This can be shown by further simplification as follows: p1 ´ αq d pDi Vq “ kp Cm VPi dt α

(29)

Substituting equation (29) into (27) we get: n p1 ´ αq ÿ d f pm, nq 1 1 d pλ1 Vq “ kp Cm V iPi ´ f pm, nq dt λ1 V α λ1 V dt

(30)

i “m

The term

řn

i “m

iPi can be represented as: n ÿ i “m

iPi “

8 ÿ

iPi ´

i “m

8 ÿ

iPi

(31)

i“n`1

Assuming, Pn = αPn´1 and Pn = (1 ´ α)αn´1 P: 8 ÿ i “m

„

 „  pn ` 1q p1 ´ αq ` α n m p1 ´ αq ` α m´1 iPi “ α P´ α P p1 ´ αq p1 ´ αq

(32)


8 of 23

And the final form of weight fraction for a semi-batch condition will be: 1 d f pm, nq “ kp Cm dt λ1

ˆ„

 „  ˙ 1 pn ` 1q p1 ´ αq ` α d pλ1 Vq m p1 ´ αq ` α αm´1 ´ αn P ´ f pm, nq α α λ1 V dt

(33)

2.1.4. Energy Balances One of the most complex features of the free radical polymerization is the exothermic nature of the reaction. Generated energy during polymerization should be removed by a coolant or dissipated to environment. Otherwise, the reactor can thermally run away. Even if run away does not occur, molar mass distribution can broaden. To model non-isothermal polymerization, energy balance should be applied to the reactant mixture in the reactor and oil in the bath. From the application of the energy conservation principle, the following equations show the energy balance for a perfectly mixed jacketed semi-batch reactor: ` ˘ ` ˘ ` ˘ d ρr Cpr VTr “ p´∆Hq pkp ` kfm qNm ˆ Cp ´ U A Tr ´ Tj ` Fm ρm Cpm ` Fi ρs Cps pTf ´ Tr q (34) dt ´ ¯ d ρj Cpj Vj Tj ` ˘ ` ˘ “ Fj ρj Cpj Tj,0 ´ Tj ` U A Tr ´ Tj (35) dt Here Tr and Tj denote the reactor and jacket temperature respectively. It was assumed that both reactor and jacket are perfectly mixed and have a constant temperature. ρr and Cpr are the average density and specific heat capacity of the reactor. Cpm , Cps and Cpj are the specific heat capacity of monomer, solvent and coolant flow which consists of water and ethylene glycol. U is the overall heat transfer coefficient and A is the heat transfer area. 2.2. Parameter Estimation Kinetic rate constants are significant parameters of a polymerization reaction which have to be determined accurately since even a slight change in them will result in considerable difference of the final polymer characteristics. The data regarding the kinetic rate constants may be obtained from literature or determined experimentally. However for some materials the properties are not available in the literature and it may not be possible to measure them through experiments due to lack of experimental facilities. Moreover, there are many criteria that affect the kinetic rate parameters such as reactor operating conditions, presence of inhibitors and purity of the materials which are different for various systems. So, proper values of the rate constants should be determined via a parameter estimation technique. This is an important prerequisite step in order to evaluate these variables and improve the model reliability for optimization and model-based control scheme development. The parameter estimation is often formulated as an optimization problem in which the estimation attempts to determine the values for the unknown parameters in order to maximize the probability that the mathematical model will predict the values obtained from the experiments. Effective solution of parameter estimation is attainable if the following criteria are met [34]: (a) The nonlinear system should be structurally identifiable which means that each set of parameter values will result in unique output trajectories. (b) Parameters which have a weak effect on the estimated measured variables and the parameters which their effect on the measured output is linearly dependent should be detected and removed from the formulation of the estimation since their effect cannot be either accurately or individually quantified. For the proposed system the parameters of the polymerization model that was introduced in section 2.1 can be represented as z(t) = [X(t), Mw (t)] which are the outputs of the parameter estimation model, u(t) = [T(t), Fm (t), Fi (t)] which are the time-varying inputs and θ the set of model parameters to be estimated which in this case are [Ad , Ap , Atd , f 0 , Tgs ]. Ad , Ap and Atd are the pre exponential factors


9 of 23

of the decomposition rate, propagation rate and termination rate respectively. The selection of these parameters are justified as the most sensitivity in conversion and weight average molar mass data is with respect to the termination and propagation rate of a polymeric chain. Proper estimation of the initiator efficiency factor is also important since it controls the effective radical concentration. Since the transition temperature for butyl acetate is not available in the literature this parameter should also be estimated. In this work the parameter estimation scheme is based on maximum likelihood criterion. The gEST function in gPROMS is used as the software to estimate the set of parameters using the data gathered from the different experimental runs. Each experiment is characterized by a set of conditions under which it is performed, which are: 1. 2. 3.

4.

The overall duration. The initial conditions which are the initial loading of initiator, solvent and monomer. The variation of the control variables. For the batch experiment temperature is the only variable, while in semi batch both temperature and flow rate of monomer and/or initiator have to be considered. The values of the time invariant parameters.

Assuming independent, normally distributed measurement errors, ijk with zero means and standard deviations, σijk this maximum likelihood goal can be captured through the following objective function: $ » ´ ¯2 fi, ’ / N Mij NV NE & . ´ ¯ i r z ´ z ÿÿ ÿ — ijk ijk N 1 ffi 2 φ “ ln p2πq ` min –ln σijk ` fl / 2 2 θ ’ σ2ijk % i “1 j “1 k “1 -

(36)

where N describes the total number of measurements taken during all the experiments, θ is the set of model parameters to be estimated which may be subjected to a given lower and upper bound, NE, NV i and NMij are respectively the total number of experiments performed, the number of variables measured in the ith experiment and the number of measurements of the jth variable in the ith experiment. σ2ijk is the variance of the kth measurement of variable j in experiment i while r zijk is the kth measured value of variable j in experiment i and zijk is the kth model-predicted value of variable j in experiment i. According to [35] the variable σ2ijk depends on the error structure of the data which can be constant (homoscedastic) or depend on the magnitude of the predicted and measured variables (heteroscedastic). If σ2ijk is fixed in the model the maximum likelihood problem is reduced into a least square criterion. If a purely heteroscedastic model applied the error has the following structure: ´ ¯γ σ2ijk “ ω2ijk r zijk

(37)

This means that as the magnitude of the measured variable increases the variance of r zijk also increases. The parameter ω2ijk and γ are determined as part of the optimization during the estimation. In this work we assume the measurement error for both conversion and weight average molecular weight in all the experiments can be described by constant variance models since the errors for both conversion and weight average molecular weight is independent of their magnitude in the measurement. The given upper and lower bounds of the variance are based on the accuracy of the measurement plant and the function gEST specifies the ω2ijk value along with X and Mw as part of the optimization. 2.3. Dynamic Optimization The objective of the dynamic optimization is to find the optimal control profile for one or more control variables and control parameters of the system that drives the process to the desired final


10 of 23

polymer property while minimizing the reaction time. The process control variables conform to their impact on the product quality and their capability for real time implementation. In this case temperature, monomer and initiator flow rate were selected as the control variables. Temperature plays a very important role in controlling the reaction kinetics which have a considerable impact on the molecular weight distribution while monomer and initiator flowrates are also a powerful means of controlling molar mass by affecting the concentration of the main feed to the reactor. The optimization of the model was performed using the gOPT function in gPROMS that applies the control vector parameterization (CVP) approach. Variation of the control variables in this case is considered as piecewise-constant, indicating the control variables remain constant at a certain value over a certain part of the time horizon before they jump discreetly to a different value over the next interval. The optimization algorithm determines the values of the controls over each interval, as well as the duration of the interval. Optimizer implements a “single-shooting” dynamic optimization algorithm consists of the following steps: 1. 2. 3. 4.

Duration of each control interval and the values during the interval are selected by the optimizer Starting from the initial condition the dynamic system is solved in order to calculate the time-variation of the states of the system Based on the solution, the values of the objective function and its sensitivity to the control variables and also the constraints are determined. The optimizer revises the choices at the first step and the procedure is repeated until the convergence to the optimum condition is achieved. For the proposed system the general optimal control problem is formulated as: min J ptf q

(38)

tf ,uptq,v

Subjected to the process model and the following constraints: x pt0 q ´ x0 “ 0

(39)

tmin ď t f ď tmax f f

(40)

umin ď u ptq ď umax

(41)

vmin ď v ptf q ď vmax

(42)

where J for the general case is defined as: ˆ J “ w1

˙2 ˆ ˙2 ˙2 ˆ ˙2 nc ˆ ÿ Mw,f f i,f tf Xf ´ 1 ` w2 ´ 1 ` w3 ´ 1 ` w4 ´1 Xt Mw,t f i,t tt

(43)

i “1

Here x0 is the initial condition of the system including the initial loading in the reactor and tf stands for the time horizon while u(t) indicates the control variables which are the temperature, monomer and initiator flow rates subjected to their lower and upper bounds. vt represents the time variant parameters being the volume of the contents of the reactor. The formulation of the objective function consists of four terms. Xf , Mw,f and f i,f are the values of the monomer conversion, molar mass and weight fraction of polymer within a chain length at the final time tf respectively and Xt , Mw,t and f i,t are their corresponding desired values. w1 –w4 are the weighting factors, determining the significance of each term in the objective function. A schematic representation of the optimization problem for the polymerization problem is given in Figure 2.

Processes 2016, 4

12 of 24

determining the significance of each term in the objective function. A schematic representation of the 11 of 23 optimization problem for the polymerization problem is given in Figure 2.


Figure 2. Schematic representation of the optimization problem in polymerization processes. Figure 2. Schematic representation of the optimization problem in polymerization processes.

3. ExperimentalSystem System 3. Experimental The The methyl methyl methacrylate methacrylatemonomer monomer stabilized stabilizedwith with 20 20 ppm ppm of of hydroquinone hydroquinone was was supplied supplied by by Sigma-Aldrich. The Azo initiator, azobisizobutyronitrile (AIBN) was supplied by Sigma-Aldrich. Butyl Sigma-Aldrich. The Azo initiator, azobisizobutyronitrile (AIBN) was supplied by Sigma-Aldrich. acetate (BA) was usedwas as solvent the polymerization reactions andreactions was supplied Sigma-Aldrich. Butyl acetate (BA) used asforsolvent for the polymerization and by was supplied by Nitrogen gas (N2) was supplied by Air Gas S.A. as an ultra-pure gas and used to keep thetoinert Sigma-Aldrich. Nitrogen gas (N2) was supplied by Air Gas S.A. as an ultra-pure gas and used keep atmosphere inside the reactor. Acetone with minimum purity of 99.5% was supplied by Sigma-Aldrich the inert atmosphere inside the reactor. Acetone with minimum purity of 99.5% was supplied by and used to clean theused reactor. Sigma-Aldrich and to clean the reactor. 3.1. Experimental Apparatus—ACOMP System 3.1. Experimental Apparatus—ACOMP System Batch and Semi Bach solution polymerization reactions were conducted in the experimental Batch and Semi Bach solution polymerization reactions were conducted in the experimental setup shown in Figure 3. The reactor shown is a 540 mL cylindrical round-bottom glass reactor setup shown in Figure 3. The reactor shown is a 540 mL cylindrical round-bottom glass reactor (Radleys, Shire Hill, Essex, United Kingdom), equipped with a double walled cooling jacket and reflux (Radleys, Shire Hill, Essex, United Kingdom), equipped with a double walled cooling jacket and condenser. Agitation of the reactor (Custom Made at Tulane University PolyRMC, New Orleans, LA, reflux condenser. Agitation of the reactor (Custom Made at Tulane University PolyRMC, New USA) was done with a U-anchor style impeller driven by an overhead stirring motor (IKA Works, Inc., Orleans, LA, USA) was done with a U-anchor style impeller driven by an overhead stirring motor Wilmington, NC, USA). The reactor temperature was controlled with the aid of a thermostatic bath and (IKA Works, Inc., Wilmington, NC, USA). The reactor temperature was controlled with the aid of a a thermocouple (Huber USA, Inc., Cary, NC, USA) connected to a data acquisition system (Brookhaven thermostatic bath and a thermocouple (Huber USA, Inc., Cary, NC, USA) connected to a data Instruments Corp., Holtsville, NY, USA). The reflux condenser (Radleys, Shire Hill, Essex, United acquisition system (Brookhaven Instruments Corp., Holtsville, NY, USA). The reflux condenser Kingdom), connected to a chilled thermostatic bath was used to prevent loss of reactants. To allow (Radleys, Shire Hill, Essex, United Kingdom), connected to a chilled thermostatic bath was used to for continuous extraction of the reactor contents, the solution was circulated from the bottom of the prevent loss of reactants. To allow for continuous extraction of the reactor contents, the solution was reactor by a Zenith 9000 series gear pump (Colfax Corporation, Annapolis Junction, MD, USA) at circulated from the bottom of the reactor by a Zenith 9000 series gear pump (Colfax Corporation, 15 mL/min. From this circulation line a small stream was pulled by a Knauer high pressure isocratic Annapolis Junction, MD, USA) at 15 mL/min. From this circulation line a small stream was pulled by pump (ICON Scientific Inc., North Potomac, MD, USA) and transferred to the first of two mixing a Knauer high pressure isocratic pump (ICON Scientific Inc., North Potomac, MD, USA) and chambers for dilution and conditioning of the polymer sample. This first stage of dilution is done under transferred to the first of two mixing chambers for dilution and conditioning of the polymer sample. atmospheric pressure to allow for degassing of trapped air in the polymer stream. The now diluted and This first stage of dilution is done under atmospheric pressure to allow for degassing of trapped air quenched polymer is again extracted by a Shimadzu LC-10ADvp isocratic pump (Shimadzu Scientific in the polymer stream. The now diluted and quenched polymer is again extracted by a Shimadzu Instruments, Houston, TX, USA) and further diluted into a static mixing tee (Idex U-466S), (IDEX LC-10ADvp isocratic pump (Shimadzu Scientific Instruments, Houston, TX, USA) and further Health & Science LLC., Oak Harbor, WA, USA). Solvent inflow for both mixing chambers was provided diluted into a static mixing tee (Idex U-466S), (IDEX Health & Science LLC., Oak Harbor, WA, USA). by multiple Shimadzu LC-10ADvp isocratic pumps (Shimadzu Scientific Instruments, Houston, TX, Solvent inflow for both mixing chambers was provided by multiple Shimadzu LC-10ADvp isocratic USA). Now that the polymer stream has been thoroughly quenched, diluted and conditioned in the pumps (Shimadzu Scientific Instruments, Houston, TX, USA). Now that the polymer stream has been ACOMP “Front End” to a concentration level sufficient for intrinsic measurements, it is flowed through thoroughly quenched, diluted and conditioned in the ACOMP “Front End” to a concentration level a series of detectors designated as the ACOMP “Detector Train”. These detectors monitor the absolute sufficient for intrinsic measurements, it is flowed through a series of detectors designated as the properties of the polymer throughout the entire polymerization process.

Processes 2016, 4

13 of 24

ACOMP “Detector Train”. These detectors monitor the absolute properties of the polymer 12 of 23 throughout the entire polymerization process.


Figure3.3. Automatic Automatic continuous continuous on-line on-line monitoring monitoring of of polymerization polymerization setup setup for for the the monitoring monitoring of of Figure methyl methacrylate solution polymerization used in this work. methyl methacrylate solution polymerization used in this work.

Thediluted dilutedsample sample stream flows through a custom built capillary single capillary viscometer The stream nownow flows through a custom built single viscometer (Custom (Custom Made at Tulane University PolyRMC, New Orleans, LA, USA), a Brookhaven Instrument Made at Tulane University PolyRMC, New Orleans, LA, USA), a Brookhaven Instrument BI-MwA BI-MwA multi-angle photometer (Brookhaven Instruments Corp., Holtsville, NY, USA) multi-angle scatteringscattering photometer (Brookhaven Instruments Corp., Holtsville, NY, USA) (operating (operating with a vertically polarized diode laser beam at 660 nm vacuum wavelength), a Shimadzu with a vertically polarized diode laser beam at 660 nm vacuum wavelength), a Shimadzu SPD-10AV SPD-10AV dualUV/visible wavelength UV/visible spectrophotometer (Shimadzu Scientific Instruments, dual wavelength spectrophotometer (Shimadzu Scientific Instruments, Houston, TX, USA) Houston, TX, USA) and a Shimadzu RID-10A differential refractometer (Shimadzu Scientific and a Shimadzu RID-10A differential refractometer (Shimadzu Scientific Instruments, Houston, TX, Instruments, Houston, TX,instrument USA). Theand signal each instrument and the output ofall the reactor USA). The signal from each the from output of the reactor thermocouple were digitized thermocouple were all digitized by A/D inputs on the BI-MwA input module, which contained 16 by A/D inputs on the BI-MwA input module, which contained 16 input channels working at 24 bit input channels working at 24 bit used resolution. The UVThe wavelengths 269 nm. of The viscometer resolution. The UV wavelengths was 269 nm. viscometerused usedwas a capillary length 10 cm used a capillary of length 10 cm and internal diameter of 0.02 in. and internal diameter of 0.02 in. 3.2. Experimental ExperimentalProcedure Procedure 3.2. Thesolvent solventininthe the reactor purged nitrogen at 30 least mintoprior to beginning the The reactor waswas purged withwith nitrogen at least min30 prior beginning the reaction. reaction. Before beginning the reaction, pure(BA) solvent was pumped at 2.0 mL/min through the Before beginning the reaction, pure solvent was(BA) pumped at 2.0 mL/min through the entire entire detectors train (UV, RI, etc.) to obtain the baseline for each instrument. After stabilization of detectors train (UV, RI, etc.) to obtain the baseline for each instrument. After stabilization of the the detector baselines pure solvent, MMA (monomer) baseline was by established by detector baselines by purebysolvent, the MMAthe (monomer) baseline was established withdrawing withdrawing from the reactor at 0.3 mL/min, and mixing with solvent at the mixing chamber. The from the reactor at 0.3 mL/min, and mixing with solvent at the mixing chamber. The flow rates of flow the rates of both thefrom purereservoir solvent from reservoir andthe that one from the reactor (containing the both pure solvent and that one from reactor (containing the reaction mixture) reaction mixture) were 2.0 respectively. and 0.2 mL/min, respectively. Providing dilution ratio used in this phase used were in 2.0this andphase 0.2 mL/min, Providing a dilution ratio ofa1:10. In all of of 1:10. In all of the experiments, the total flow rate was set at 2.0 mL/min and the diluted solution the experiments, the total flow rate was set at 2.0 mL/min and the diluted solution always reached the always reached detector train of 25 °C. After baseline foradded MMA the detector train at athe temperature of 25at˝aC.temperature After baseline stabilization for MMAstabilization the AIBN was to AIBN was to initiate the reaction. initiate the added reaction. 4. Results 4. Resultsand and Discussion Discussion In this this section, section, model model predictions predictions are arecompared comparedwith with the the literature literaturedata dataand andexperiments experimentswhich which In have been been performed performed using using ACOMP ACOMP system. in terms terms of of the the conversion conversion have system. The The results results are are illustrated illustrated in historyand andthe the evolution evolution of of molar mass and and molar molar mass mass distribution. distribution. In In the the first first case casethe the proposed proposed history molar mass model is is tested tested under under aa different different system systemof of initiator initiatorand andsolvent. solvent. The The model model is is then then verified verified against against model experimental data where the polymerization of methyl methacrylate using butyl acetate as solvent experimental data where the polymerization of methyl methacrylate using butyl acetate as solvent and and AIBN the initiator is considered. AIBN as theasinitiator is considered. 4.1. 4.1. Validation ValidationUsing Using Literature LiteratureData Data The The experimental experimentaldata dataprovided providedby byCrowley Crowley et et al. al. [31] [31]is is used usedto to investigates investigatesthe theapplicability applicabilityof of the the proposed proposedframework. framework.Batch Batchfree freeradical radicalsolution solutionpolymerization polymerizationofofmethyl methylmethacrylate methacrylatehas hasbeen been carried out in a 4 L jacketed stirred tank reactor which was initially charged with 500 mL of MMA, 500 mL of ethyl acetate solvent, and 9 g of 2,2¢-azobis(2-methylbutanenitrile) initiator. The reactor was

Processes 2016, 4 Processes 2016, 4

14 of 24 14 of 24

carried Processes out 2016,in 4, 5a 4 L jacketed stirred tank reactor which was initially charged with 500 mL of MMA, 13 of 23 carried in aacetate 4 L jacketed stirred which was initially charged initiator. with 500 The mL of MMA, 500 mL ofout ethyl solvent, and 9tank g of reactor 2,2¢-azobis(2-methylbutanenitrile) reactor 500 mL of ethyl acetate solvent, and 9 g of 2,2¢-azobis(2-methylbutanenitrile) initiator. The reactor was heated to 65 °C, regulated at that temperature to 27% monomer conversion, and then the was heated to °C, regulated at temperature to 27% conversion, monomer conversion, and then the heated to 65 ˝was C, 65 regulated at to that to 27% monomer andweight then thedistribution. temperature temperature decreased 50temperature °Cthat to intentionally broaden the molecular ˝ C to intentionally temperature was decreased to 50 °C to intentionally broaden the molecular weight distribution. was decreased to 50 broaden the molecular weight distribution. Results from the Results from the paper and simulation from the model are represented below for the conversion and Results from thedistribution paper andthe simulation model are below for the conversion and paper and simulation from model arefrom represented below for conversion and weight molecular weight at the end of thethe batch. As canrepresented be the observed in Figure 4molecular they are in quite molecular weight distribution at the end of the batch. As can be observed in Figure 4 they are in quite distribution at the end of the batch. As can be observed in Figure 4 they are in quite good agreement. good agreement. good agreement.

(a) (a)

(b) (b)

Figure 4. Comparison between model simulations and literature results. (a) Panel-Conversion profile. Figure 4.4.Comparison between simulations and Figure Comparison betweenmodel model andliterature literatureresults. results. (a) (a) Panel-Conversion Panel-Conversion profile. profile. (b) Panel-Molar mass distribution at the simulations end of the batch. (b) Panel-Molar mass distribution at the end of the batch. (b) Panel-Molar mass distribution at the end of the batch.

(a) (a)

(b) (b)

Figure 5. Optimization results: (a) Temperature profiles; (b) Molar mass distribution. Figure 5. Optimization results: (a) Temperature profiles; (b) Molar mass distribution. Figure 5. Optimization results: (a) Temperature profiles; (b) Molar mass distribution.

Data from Crowley et al. [31] was also utilized for the optimal batch operation. Based on the Datamodel from Crowley et al.to [31] was the alsooptimal utilizedtrajectory for the optimal batch operation. onthe the developed it is possible obtain of the temperature whichBased lead to Data from Crowley et al. [31] was also utilized for the optimal batch operation. Based on the developed model it is possible to obtain the optimal trajectory of the temperature which lead to the desired molar mass distribution of polymers. The main problem in the batch optimization is the final developed model it is possible to obtain the optimal trajectory of the temperature which lead to the desired molar mass distribution of polymers. The main problem the batch optimization is the final time which is not specified. The proposed solution is to set all the in kinetic equations with respect to desired molar mass distribution ofproposed polymers. The main problem in the batchequations optimization isrespect the final time which is not specified. The solution is to set all the kinetic with conversion since its final value has already been specified. The solution of the optimization is ato time which is not its specified. The has proposed solution is to set all the kineticof equations with respect conversion since final value already been specified. the optimization temperature profile with respect to conversion. The profile can The thensolution be implemented and the stateisofa to conversion since its final value has already been specified. The solution of the optimization is of a temperature profile with respect to conversion. The profile can then be implemented and the state


14 of 23

temperature profile with respect to conversion. The profile can then be implemented and the state of the system is monitored with respect to time. The objective now is to reach this final distribution by manipulating temperature before we reach the specified monomer conversion which is 0.7 in this case. There are two criteria that affect the results of the optimization. First, the number of intervals for the monomer conversion and second the initial guess for the temperature profile. Here the results obtained by manipulating the number of intervals are presented. Results for 1 and 5 intervals are provided in Figure 5, which illustrates the MMD for each case, as compared with the target. The results are in good agreement with those obtained in the original paper. 4.2. Experimental Validation for Batch and Semi-Batch Free Radical Polymerization of MMA Using Butyl Acetate as Solvent and AIBN Initiator The experimental set-up discussed above was employed in this case to fully validate the proposed strategy for both batch and semibatch operation; Up to now previous works in the literature had been based on infrequent and incomplete sets of data and/or limited to batch operation. In this work model validations and parameter estimation have been carried out using ACOMP data. The capability of the model to properly describe the polymerization system has been investigated by doing a number of experiments in both semi-batch and batch mode using different trajectories for temperature and initiator and monomer flow rate. Conversion, weight average molar mass as well as molar mass distribution trajectories along the whole process were evaluated. In an initial step, using a set of the experimental data, parameter estimation was performed for the main kinetic parameters. The optimal values of the estimated parameters as well as the uncertainty of the parameter represented as 95% confidence interval (CI) are shown in Table 2. Here, the contours correspond to a confidence level of 95%. In other words, there is a probability of 95% that the true values of the parameter pair fall within this ellipsoidal confidence region that is centered in the parameter estimates. In addition to confidence intervals the correlation matrix is also represented in Table 3 as a 5 ˆ 5 lower triangular matrix (the upper triangular matrix is identical to the lower one). The most pronounced correlations between the parameters are shown in bold. Table 2. Original and estimated value of the kinetic rate parameters for the free radical polymerization of MMA (first iteration). Parameter

Description

Confidence Interval

Original Value

Estimated Value

90%

95%

99%

95% t-value

Standard Deviation

Ad

Decomposition (1/min)

1.58 ˆ 1015

1.37 ˆ 1015

1.25 ˆ 1014

1.49 ˆ 1014

1.96 ˆ 1014

9.19

7.60 ˆ 1013

Ap

Propagation (m3 /mol¨min)

4.2 ˆ 105

9 ˆ 105

5.23 ˆ 104

6.23 ˆ 104

8.20 ˆ 104

14.43

3.17 ˆ 104

Atd

Termination [m3 /mol¨min]

1.06 ˆ 108

4.56 ˆ 108

5.97 ˆ 107

7.12 ˆ 107

9.37 ˆ 107

6.40

3.63 ˆ 107

f0

Initial Initiator Efficiency

0.58

0.57

0.048

0.057

0.076

9.84

0.029

Ts

Solvent Transition Temperature (K)

181

142.61

0.539

0.6431

0.84

221.7

0.327

Table 3. Correlation matrix for the estimated parameters of the system. Estimated Parameters

Ad

Ap

At

f0

Ts

Ad Ap At f0 Ts

1 0.117 0.111 –0.988 0.104

1 0.988 0.038 0.179

1 0.043 0.233

1 –0.070

1

Processes 2016, 4, 5 Processes 2016, 4

15 of 23 16 of 24

Most of of the parametersobtained obtainedhave havenarrow narrowconfidence confidenceintervals intervalsindicating indicatingthat thatthe the Most the estimated estimated parameters number of of measurements measurements performed performed for for the the parameter parameter estimation estimationwere were sufficient. sufficient. The The normalized normalized number covariance matrix matrix shows shows that that although although aa few covariance few parameters parameters are are quite quite correlated, correlated, most most parameters parameters estimatedininthe the optimization weakly correlated and therefore are for suitable being estimated optimization are are onlyonly weakly correlated and therefore are suitable being for estimated estimated simultaneously. Larger correlation coefficients found between the propagation rate and simultaneously. Larger correlation coefficients are foundare between the propagation rate and the the termination rate as as well as initial initiator efficiency and decomposition rate asinshown termination rate as well initial initiator efficiency and decomposition rate as shown Figure in 6. Figure 6. Likely, any in oneparameters of these parameters could be compensated change the Likely, any change inchange one of these could be compensated by a changebyinathe otherinones. other ones. For thebetween coefficient between 𝐴𝐴td is 0.94 indicating a strong correlation For example, theexample, coefficient Ap and Atd is𝐴𝐴0.94 indicating a strong correlation between them p and between them and making to find afor unique for Unique these parameters. Unique and making it difficult to findit adifficult unique estimate these estimate parameters. parameter estimate parameter estimate meanshave thatan theacceptably parameters an acceptably low to any of theaother means that the parameters lowhave correlation to any of thecorrelation other parameters and low parametersinterval. and a low confidence Thus, in spite ofmentioned the large covariance mentioned above, a confidence Thus, in spiteinterval. of the large covariance above, a consistent estimation consistent possible because of estimated the true value of the estimated parameters is possible estimation because of is the true value of the parameters are located within a are verylocated small within a very small confidence reducing their uncertainty. However, theare confidence ellipsoids confidence bands reducing theirbands uncertainty. However, the confidence ellipsoids large including in are large including most cases negative Therefore, a second iteration was performed by most cases negativein numbers. Therefore, a numbers. second iteration was performed by eliminating two of the and 𝐴𝐴 and fixing their values to the estimated eliminating two of theAcorrelated parameters 𝐴𝐴 correlated parameters and A and fixing their values to the estimated ones in the first iteration. d td d td onesoptimal in the first iteration. The optimal values ofasthe estimated parameters as well as the uncertainty The values of the estimated parameters well as the uncertainty of the parameter represented of 95% the parameter 95% confidence interval (CI) are shown in Table 4. as confidencerepresented interval (CI)asare shown in Table 4.

Figure 6. Confidence ellipsoids for 𝐴𝐴 d –𝑓𝑓0 and 𝐴𝐴 p –𝐴𝐴 t. Figure 6. Confidence ellipsoids for Ad –f 0 and Ap –At .

Table 4. Original and estimated value of the kinetic rate parameters (second iteration) for the free Table 4. Original and estimated value of the kinetic rate parameters (second iteration) for the free radical polymerization of MMA. radical polymerization of MMA. Parameter Parameter Ap

𝐴𝐴p

f0

𝑓𝑓0

Ts

𝑇𝑇s

Description Description

Original

Propagation Rate Rate Propagation 3 ˆ 105 105 3 × 105 8.5 ˆ 8.5 × 105 (m3 /mol¨min)

(m 3/mol∙min) Initial Initiator Initial Initiator 0.58 Efficiency 0.58 Efficiency Solvent Transition 142 Solvent Temperature (K) Transition 142 Temperature (K)

Confidence Interval

Estimated

Original Estimated ValueValue ValueValue

0.56

149.94

0.56

149.94

95% t-

Confidence Interval

95% t-value value

Standard

Standard Deviation Deviation

90% 90%

95% 95%

99%99%

2547

3035

3035

3993

280.1

1546

0.0013

2547

3993

280.1

1546

0.001166

0.0013

0.00182

403.2

0.00073

0.3906

0.465

0.6123

322.2

0.237

0.001166 0.3906

0.465

0.00182 0.6123

403.2

322.2

0.00073 0.237

For purposes of illustration the confidence regions for the parameter pairs estimated are shown in Figure 7.purposes None of the confidence ellipsoids cross regions either the or yparameter axes. Thus, no parameter has a For of illustration the confidence forxthe pairs estimatedpair are shown parameter value equal to zero. The confidence ellipsoids also show small negative correlation between in Figure 7. None of the confidence ellipsoids cross either the x or y axes. Thus, no parameter pair has propagation and corresponding glass transition temperature for small the solvent. a parameterconstant value equal to zero. The confidence ellipsoids also show negative correlation

between propagation constant and corresponding glass transition temperature for the solvent.


16 of 23

Processes 2016, 4

17 of 24

Processes 2016, 4

17 of 24

Figure Confidenceellipsoids ellipsoidsfor for final final estimated estimated parameters. Figure 7. 7. Confidence parameters. Figure 7. Confidence ellipsoids for final estimated parameters.

Figure 8 shows the conversion and molar mass profiles for two batch conditions with constant Figure 8 shows the conversion and molar massprofiles profiles for two batch conditions with constant Figure temperature. 8 shows the conversion molar mass forestimation two batch conditions withimproves constant the and varying As can beand observed, the parameter significantly and and varying temperature. As be theparameter parameter estimation significantly improves the varying temperature. Ascan can beobserved, observed, the estimation significantly improves the the results. The simulation results after the parameter estimation have an excellent agreement with results. TheThe simulation results after estimationhave have excellent agreement with the results. simulation results afterthe theparameter parameter estimation anan excellent agreement with the experimental data for both conditions, indicating the adjusted model has good predictive capabilities experimental datadata forfor both conditions, modelhas hasgood good predictive capabilities experimental both conditions,indicating indicatingthe the adjusted adjusted model predictive capabilities for the proposed system. forproposed the proposed system. for the system.

(a) (a)

(b)

(b)

Figure 8. Comparison between experimental data and simulation with original and estimated Figure 8. 8.Comparison between experimental data simulation with original originaland andestimated estimated Figure Comparison between experimental data and andexperiment. simulation with parameters: (a) Isothermal experiment, (b) Non-isothermal

parameters: (a)(a) Isothermal experiment. parameters: Isothermalexperiment; experiment,(b) (b)Non-isothermal Non-isothermal experiment. Next, the adjusted model is embedded into the optimization environment to investigate alternative optimal operational policies for target Twoenvironment objective functions were Next, thethe adjusted model is embedded intofinal the optimization environment to investigate alternative Next, adjusted model is embedded into the products. optimization to investigate formulated. One is tooperational determine the optimal ofobjective the control input that minimize the alternative optimal policies fortrajectories final target products. Two values objective functions were optimal operational policies for final target products. Two functions were formulated. One is reaction time while the product qualities reach the specification at the end of the process and the formulated. One is to determine theof optimal trajectories of the control input values that minimize the to determine the optimal trajectories the control input values that minimize the reaction time while other will only consider the properties atthe thespecification end of the process enough time toand the time while the product qualities reach at and thegiven end the will process the thereaction product qualities reach the product specification at the end of the process the of other only consider

will properties only consider the end product properties the end of thetime process given enoughThe time to the theother product at the of the process at given enough to the reaction. optimizer iteratively computes the sequence of reactor temperature, monomer and initiator flow rates which


17 of 23

Processes 2016, 4 Processes 2016, 4

18 of 24 18 of 24

will yield theThe bestoptimizer match between final conversion, molar masstemperature, and molar monomer mass distribution reaction. iterativelythe computes the sequence of reactor and reaction. The optimizer iteratively computes the sequence of reactor temperature, monomer and and their corresponding target values. Figure 9 presents a snapshot of the three selected iterations. initiator flow rates which will yield the best match between the final conversion, molar mass and initiator rates which will yield thecorresponding best between the final conversion, molar mass and The graph on the left-hand side shows thematch conversion and the9 right diagram represents molarflow mass distribution and their targetprofile values. Figure presents a snapshot of the the molar mass distribution and their corresponding target values. Figure 9 presents a snapshot of the three selected iterations. graph Since on thethe left-hand sidefor shows theiterations conversionare profile and the right calculated objective functionThe values. solution early not optimal the final three selected iterations. The graph the left-hand side shows profile and the right are diagram represents the calculated function values. Since the solution for early iterations objective function is high and thereonisobjective a large discrepancy inthe theconversion final conversion. Simulation results diagram represents the calculated objectiveisfunction values. Since thediscrepancy solution for in early iterations are not optimal the final objective function high and there is a large the final conversion. of optimal profiles at final iteration are shown in Figure 10 for the three control variables and as can not Simulation optimal the results final objective function is high anditeration there is aare large discrepancy optimal profiles at final shown in Figurein10the forfinal the conversion. three control be observed there is anofexcellent agreement between the final values and the target suggesting the Simulation optimal profilesthere at final iteration areagreement shown in Figure 10 the for the control variablesresults and asofcan be observed is an excellent between finalthree values and the advantage of using and monomer flow withagreement temperature to control not onlyand the the conversion variables as canreagent be there is anreagent excellent the final values target and suggesting theobserved advantage of using and monomerbetween flow with temperature to control not and weight average molecular weight but also the complete distribution. It should be noted target suggesting the advantage of using reagent and monomer control not that only the conversion and weight average molecular weightflow but with also temperature the completetodistribution. It the optimizer, CVP-S (Control Vector ParameterizationSingle Shooting), used by gPROMS may onlyshould the conversion andthe weight average molecular weight but also the complete distribution. be noted that optimizer, CVP-S (Control Vector ParameterizationSingle Shooting),Itused find should be noted may that the CVP-S one. (Control Vector ParameterizationSingle used the local minima rather than the minima global So, thethe solution is sensitive to theShooting), initial point by gPROMS findoptimizer, the local rather than global one. So, the solution is sensitive to and the it is by gPROMS may find the local minima rather than the global So, to thecapture solutionthe is sensitive to the the necessary properly thetooptimization problem inone. order global in case initialtopoint and it initialize is necessary properly initialize the optimization problem in order tominima capture point andexistence. it is to properly initialize the optimization problem in order to capture the global minima innecessary case of local minima existence. ofinitial local minima global minima in case of local minima existence.

(a)

(a)

(b)

(b)

Figure 9. Conversion (a) and objective function (b) profiles at three different iterations.

Figure 9. Conversion Conversion(a)(a) and objective function (b) profiles at different three different iterations. Figure 9. and objective function (b) profiles at three iterations.

(a)

(a)

(b) (b) Figure 10. Simulation results of the optimal trajectories considering time in the objective function: Figure 10. Simulation results of the optimal trajectories considering time in the objective function: (a)10. Input (manipulated) variables; Controlled variables (targets). Figure Simulation results of the(b) optimal trajectories considering time in the objective function: (a) Input (manipulated) variables; (b) Controlled variables (targets).

(a) Input (manipulated) variables; (b) Controlled variables (targets).

Processes 2016, 4 Processes 2016, 4, 5

19 of 24 18 of 23

To demonstrate the feasibility of the proposed optimization strategy the second optimization algorithm for the objective has validatedoptimization also experimentally data from ACOMP. To demonstrate the function feasibility of been the proposed strategyusing the second optimization Table 5 provides the values of the different parameters and the constraints used in this experiment. algorithm for the objective function has been validated also experimentally using data from ACOMP. Temperature and flow used based on the capacity of the pumpused andin jacket while the Table 5 provides the constraints values of theare different parameters and the constraints this experiment. minimum volume constraint is the minimum necessary volume for the sensors to have a good Temperature and flow constraints are used based on the capacity of the pump and jacket while estimation of the reactor condition. Figure 11 shows the resulting optimal trajectories of the the minimum volume constraint is the minimum necessary volume for the sensors to haveinput a good variables (temperature, monomer andFigure initiator the validation results in terms of the estimation of the reactor condition. 11 flows) shows with the resulting optimal trajectories of the input model predictions (targets) and the experimental data when the obtained inputs trajectories are variables (temperature, monomer and initiator flows) with the validation results in terms of the model applied into the experimental systems. The complete distribution has been shown at the final of predictions (targets) and the experimental data when the obtained inputs trajectories aretime applied theinto experiment. The maximum deviations from the model simulation is in the order of 0.03 for the experimental systems. The complete distribution has been shown at the final time of the conversion andThe 5 kg/mol for the molar mass. final distribution also a very agreement experiment. maximum deviations from The the model simulation isisin thein order of good 0.03 for conversion with the model predictions showing the feasibility of the proposed scheme. and 5 kg/mol for the molar mass. The final distribution is also in a very good agreement with the model predictions showing the feasibility of the proposed scheme.

Table 5. Values of optimization constraints parameters. Table 5. Values of optimization constraints parameters.

Variable 𝑁𝑁m Variable 𝑁𝑁 Nsm 𝑁𝑁 Ni s Ni 𝐹𝐹max max 𝐹𝐹Fmin Fmin 𝑇𝑇max Tmax 𝑇𝑇Tmin min 𝑉𝑉Vmax max 𝑉𝑉Vmin min

Value

0.5Value 0.5 0.5 0.01 0.5 5 0.01 0 5 0 70 70 50 50 500 500 100 100

Unit

mol Unit mol mol mol mol mL/minmol mL/min mL/min mL/min °C ˝ C °C ˝ C mL mL mL mL

(a)

(b) Figure 11. Validation of optimal runs: (a) Input (manipulated) variables; (b) Controlled variables (targets). Figure 11. Validation of optimal runs: (a) Input (manipulated) variables; (b) Controlled variables (targets).

To analyze the controllability of the system under feedback control and as a preliminary step towards the on-line feedback control implementation, a number of simulations combined with


19 of 23

To analyze the controllability of the system under feedback control and as a preliminary step towards the on-line feedback control implementation, a number of simulations combined with experimental information were performed. In this study, the experimental data from the optimal Processes 20 Processes2016, 2016,44 20of of 24 24 semi-batch operation for the corresponding controlled variable was used as a set-point in a simulated experimental were In study, the data experimental information were performed. In this this study,profiles the experimental experimental data from from the optimal feedback scheme.information In this way theperformed. experimental optimal for conversion (inthe theoptimal single loop semi-batch operation for the corresponding controlled variable was used as aaset-point in aasimulated semi-batch operation for the corresponding controlled variable was used as set-point in simulated case) and experimental optimal profiles of both conversion and molar mass flow (in the multi-loop feedback scheme. this way the profiles for conversion the single loop scheme.In In theexperimental experimentaloptimal optimal profiles conversion(in (in theboth singletemperature loopcase) case) case) feedback were provided to this the way simulator/controller to adjust thefor temperature and and experimental optimal profiles of both conversion and molar mass flow (in the multi-loop case) and experimental optimal profiles of both conversion and molar mass flow (in the multi-loop case) and initiator flow to achieve the targets. The adjusted input trajectories are then compared to the were were provided provided to to the the simulator/controller simulator/controller to to adjust adjust the the temperature temperature and and both both temperature temperature and and experimental ones showing the necessary adjustments needed for the control system to carethe for the initiator initiator flow flow to to achieve achieve the the targets. targets. The The adjusted adjusted input input trajectories trajectories are are then then compared compared to to the model uncertainties. The objective in this part is to propose a straightforward modification to the experimental experimental ones ones showing showingthe thenecessary necessaryadjustments adjustments needed neededfor for the thecontrol controlsystem systemto tocare carefor forthe the open-loop recipe described the process to meetmodification the optimalto condition model uncertainties. The objective in part is propose aa straightforward the modeloptimal uncertainties. The objectiveabove in this thisthat partwill is to toallow propose straightforward modification to the open-loop optimal recipe described above that will allow the process to meet the optimal condition under non-ideal process circumstances. open-loop optimal recipe described above that will allow the process to meet the optimal condition under non-ideal process circumstances. The block diagram of the proposed structure is shown in Figure 12. The three possible manipulated under non-ideal process circumstances. The block diagram of structure shown in The three possible The block of the the proposed proposed structure shown flow in Figure Figure 12.the The threeloop possible variables in this casediagram are the temperature, monomer andisisinitiator rate. 12. In single approach, manipulated variables in this case are the temperature, monomer and initiator flow rate. In the in this variables case are the temperature, and initiator flow rate. thesingle single two ofmanipulated the possiblevariables manipulated still follow themonomer same optimal recipe while theIn other variable loop approach, two of the possible manipulated variables still follow the same optimal recipe while loop approach, two of the possible manipulated variables still follow the same optimal recipe while changes based on a PI-like control algorithm in order to follow exactly the set point trajectory. In this the changes based on control algorithm in order to the set theother othervariable variable changes basedas onathe aPI-like PI-like control algorithm tofollow followexactly exactly setpoint point case trajectory. conversion has been selected control variable andinitsorder controllability withthe respect to the In this case conversion has been selected as the control variable and its controllability trajectory. In this case conversion has been selected as the control variable and its controllabilitywith with reactor temperature is investigated. The temperature adjusted using a cascade control system which respect respectto tothe thereactor reactortemperature temperatureis is investigated. investigated.The Thetemperature temperatureadjusted adjustedusing usingaacascade cascadecontrol control has been represented schematically inschematically Figure 13. The approach followed here is to start the operation system which has been represented in Figure 13. The approach followed system which has been represented schematically in Figure 13. The approach followedhere hereisisto tostart start in anthe open-loop fashion and switch to feedback control during the batch operation. The switching operation in an open-loop fashion and switch to feedback control during the batch operation. The the operation in an open-loop fashion and switch to feedback control during the batch operation. The switching time is around 20 min for the single loop and 60 min for the multi loop controller. Figure time is around time 20 min for the20 single loop 60 min multi controller. Figure 14Figure shows the switching is around min for theand single loop for andthe 60 min forloop the multi loop controller. 14 the performance of closed loop system. As seen of performance closed loop system. can be seen good of control the conversion through the 14 shows showsof thethe performance of the the closedAs loop system. As can can be becontrol seen good good control ofthe theconversion conversion through the batch operation was attained by slight modifications of the open-loop optimal thewas batch operation attained by slight modifications of the open-loop optimaltrajectories trajectories batchthrough operation attained bywas slight modifications of the open-loop optimal trajectories as predicted as predicted from the optimization step. However the molar mass is still out of specifications with as predicted from the optimization step. However the molar mass is still out of specifications withto the from the optimization step. However the molar mass is still out of specifications with respect respect to the desired final value. respect to the desired final value. desired final value.

Figure Schematic apparatus and sensors control strategy. Figure 12. 12. Schematic ofof apparatus forthe theproposed proposed control strategy. Figure 12. Schematic of apparatusand andsensors sensorsfor for the proposed control strategy.

Figure 13. Schematic of cascade control for conversion.

Figure Schematicofofcascade cascade control Figure 13.13. Schematic controlfor forconversion. conversion.

Processes 2016, 4, 5 Processes 2016, 4 Processes 2016, 4

20 of 23 21 of 24 21 of 24

Figure 14. loop controller. Figure 14. Closed-loop Closed-loop results using single loop Figure 14. Closed-loop results using single loop controller.

In was again again controlled controlledusing usingthe thecascade cascadeconfiguration configuration In the the multi-loop multi-loop approach approach the the conversion conversion was In the multi-loop approach the conversion was again controlled using the cascade configuration as described above but an additional loop was added by adjusting the initiator flow to as described above but an additional loop was added by adjusting the initiator flow to control control the the as described above but an additional loop was added by adjusting the initiator flow to control the molar mass. molar mass trajectories follow exactly the molar mass. Figure Figure15 15illustrates illustrateshow howboth bothconversion conversionand and molar mass trajectories follow exactly molar mass. Figure 15 illustrates how both conversion and molar mass trajectories follow exactly the desired values as indicated in the plot. One should notice however that small corrections in the the desired values as indicated in the plot. One should notice however that small corrections in desired values as indicated in the plot. One should notice however that small corrections in the molecular weight requires a large control action (initiator the molecular weight requires a large control action (initiatorflow) flow)indicating indicatinglow lowsensitivity sensitivity of of the the molecular weight requires a large control action (initiator flow) indicating low sensitivity of the manipulatedvariable variablewith withrespect respecttotothe thecontrol controlvariable. variable.Also, Also,the theinteraction interaction between thetwo twoloops loops manipulated between the manipulated variable with respect to the control variable. Also, the interaction between the two loops can be be appreciated appreciatedsince sincethe thetemperature temperatureand andconversion conversion profiles slightly modified with respect can profiles areare slightly modified with respect to can be appreciated since the temperature and conversion profiles are slightly modified with respect to the single loop case. Specially during first action of the initiator controller. the single loop case. Specially during thethe first action of the initiator controller. to the single loop case. Specially during the first action of the initiator controller.

Figure 15. Closed-loop results for multi-loop controllers. Figure Figure 15. 15. Closed-loop Closed-loop results results for for multi-loop multi-loop controllers. controllers.


21 of 23

5. Conclusions Formulation and implementation of a model-based framework for prediction and understanding of free radical polymerization processes has been dealt in this paper. The proposed framework is based on the recent development in characterizing molecular weight distribution (MWD) in polymerization processes combined with a state-of–the art experimental tools for on-line monitoring of polymeric properties towards the optimal operations of this type of systems. The method of finite molecular weight moments has been proposed to represent MWD. This allows computation of the complete chain length distribution instead of molecular weight averages. The model presented was modified for application in semi batch processes. The parameters of the kinetic model were then identified and estimated for the case study of MMA free radical polymerization. Dynamic optimization of the system using the CVP technique was undertaken to yield a product with desired polymer molecular weight characteristics in terms of monomer conversion, weight average molecular weight and the entire desired molecular weight distribution. The model-based optimal policy was validated experimentally and fair to good results were achieved. Preliminary feedback control algorithms were implemented and tested through simulations to achieve a desired conversion and weight average molecular weight of the system. They included single-loop and multi-loop Proportinal-Integral control strategies. Building on these results, current efforts focus on developing the potential and expanding the capabilities of this new on-line measuring method and modelling approach to fully characterize and control the dynamics involved in polymerization systems. Incorporating advanced control policies with an on line execution level controller and application to other polymerization processes will be the subject of the forthcoming paper. Acknowledgments: The authors acknowledge support from United States Department of Energy, Advanced Manufacturing Office, DE-EE0005776 and the National Science Foundation under the NSF EPSCoR Cooperative Agreement No. EPS-1430280 with additional support from the Louisiana Board of Regents. Author Contributions: Navid Ghadipasha and Aryan Geraili conceived the programming and design algorithm for modelling, simulation, estimation, optimization and control part under Jose A. Romagnoli’s supervision. The experiments were designed and performed by Michael F. Drenski and Carlos A. Castor under Wayne F. Reed’s Supervision. The section “Experimental system” was written by Michael F. Drenski. Navid Ghadipasha wrote the other sections while everyone supervised the writing process. Conflicts of Interest: The authors declare no conflict of interest.

References 1.

2. 3. 4.

5. 6. 7.

8.

Wu, T.; Yu, L.; Cao, Y.; Yang, F.; Xiang, M. Effect of molecular weight distribution on rheological, crystallization and mechanical properties of polyethylene-100 pipe resins. J. Polym. Res. 2013, 20, 1–10. [CrossRef] Schimmel, K.H.; Heinrich, G. The influence of the molecular-weight distribution of network chains on the mechanical-properties of polymer networks. Colloid Polym. Sci. 1991, 269, 1003–1012. [CrossRef] Malekmotiei, L.; Samadi-Dooki, A.; Voyiadjis, G.Z. Nanoindentation study of yielding and plasticity of poly(methyl methacrylate). Macromolecules 2015, 48, 5348–5357. [CrossRef] Guyot, A.; Guillot, J.; Pichot, C.; Guerrero, L.R. New design for production of constant composition co-polymers in emulsion polymerization—Comparison with co-polymers produced in batch. Abstr. Pap. Am. Chem. S 1980, 180, 131–ORPL. Dimitratos, J.; Georgakis, C.; Elaasser, M.S.; Klein, A. Dynamic modeling and state estimation for an emulsion copolymerization reactor. Comput. Chem. Eng. 1989, 13, 21–33. [CrossRef] Dimitratos, J.; Georgakis, C.; Elaasser, M.; Klein, A. An experimental-study of adaptive kalman filtering in emulsion copolymerization. Chem. Eng. Sci. 1991, 46, 3203–3218. [CrossRef] Hammouri, H.; McKenna, T.F.; Othman, S. Applications of nonlinear observers and control: Improving productivity and control of free radical solution copolymerization. Ind. Eng. Chem. Res. 1999, 38, 4815–4824. [CrossRef] Dochain, D.; Pauss, A. Online estimation of microbial specific growth-rates—An illustrative case-study. Can. J. Chem. Eng. 1988, 66, 626–631. [CrossRef]


9. 10. 11. 12. 13.

14. 15.

16. 17. 18. 19. 20. 21. 22. 23.

24. 25.

26. 27. 28. 29. 30. 31. 32.

22 of 23

Kozub, D.J.; Macgregor, J.F. State estimation for semibatch polymerization reactors. Chem. Eng. Sci. 1992, 47, 1047–1062. [CrossRef] Mutha, R.K.; Cluett, W.R.; Penlidis, A. On-line nonlinear model-based estimation and control of a polymer reactor. Aiche. J. 1997, 43, 3042–3058. [CrossRef] Mutha, R.K.; Cluett, W.R.; Penlidis, A. A new multirate-measurement-based estimator: Emulsion copolymerization batch reactor case study. Ind. Eng. Chem. Res. 1997, 36, 1036–1047. [CrossRef] Kravaris, C.; Wright, R.A.; Carrier, J.F. Nonlinear controllers for trajectory tracking in batch processes. Comput. Chem. Eng. 1989, 13, 73–82. [CrossRef] Alhamad, B.; Romagnoli, J.A.; Gomes, V.G. On-line multi-variable predictive control of molar mass and particle size distributions in free-radical emulsion copolymerization. Chem. Eng. Sci. 2005, 60, 6596–6606. [CrossRef] Garcia, C.E.; Morari, M. Internal model control.1. A unifying review and some new results. Ind. Eng. Chem. Proc. Dd. 1982, 21, 308–323. [CrossRef] Park, M.J.; Rhee, H.K. Control of copolymer properties in a semibatch methyl methacrylate/methyl acrylate copolymerization reactor by using a learning-based nonlinear model predictive controller. Ind. Eng. Chem. Res. 2004, 43, 2736–2746. [CrossRef] Gattu, G.; Zafiriou, E. Nonlinear quadratic dynamic matrix control with state estimation. Ind. Eng. Chem. Res. 1992, 31, 1096–1104. [CrossRef] Lee, J.H.; Ricker, N.L. Extended kalman filter based nonlinear model-predictive control. Ind. Eng. Chem. Res. 1994, 33, 1530–1541. [CrossRef] Henson, M.A. Nonlinear model predictive control: Current status and future directions. Comput. Chem. Eng. 1998, 23, 187–202. [CrossRef] Schork, F.J.; Deshpande, P.B.; Leffew, W.K. Control of polymerization reactors; CRC Press: Boca Raton, FL, USA, 1993; pp. 101–104. Crowley, T.J.; Choi, K.Y. Experimental studies on optimal molecular weight distribution control in a batch-free radical polymerization process. Chem. Eng. Sci. 1998, 53, 2769–2790. [CrossRef] Ellis, M.F.; Taylor, T.W.; Jensen, K.F. Online molecular-weight distribution estimation and control in batch polymerization. Aiche. J. 1994, 40, 445–462. [CrossRef] Congalidis, J.P.; Richards, J.R.; Ray, W.H. Feedforward and feedback-control of a solution copolymerization reactor. Aiche. J. 1989, 35, 891–907. [CrossRef] Adebekun, D.K.; Schork, F.J. Continuous solution polymerization reactor control 2. Estimation and nonlinear reference control during methyl-methacrylate polymerization. Ind. Eng. Chem. Res. 1989, 28, 1846–1861. [CrossRef] Florenzano, F.H.; Strelitzki, R.; Reed, W.F. Absolute, on-line monitoring of molar mass during polymerization reactions. Macromolecules 1998, 31, 7226–7238. [CrossRef] Giz, A.; Catalgil-Giz, H.; Alb, A.; Brousseau, J.L.; Reed, W.F. Kinetics and mechanisms of acrylamide polymerization from absolute, online monitoring of polymerization reaction. Macromolecules 2001, 34, 1180–1191. [CrossRef] Reed, W.F. Automated continuous online monitoring of polymerization reactions (acomp) and related techniques. Anal. Chem. 2013. [CrossRef] Baillagou, P.E.; Soong, D.S. Major factors contributing to the nonlinear kinetics of free-radical polymerization. Chem. Eng. Sci. 1985, 40, 75–86. [CrossRef] Pinto, J.C.; Ray, W.H. The dynamic behavior of continuous solution polymerization reactors 7. Experimental-study of a copolymerization reactor. Chem. Eng. Sci. 1995, 50, 715–736. [CrossRef] Ray, W.H. Mathematical modeling of polymerization reactors. J. Macromol. Sci. R M C 1972, 8, 1–56. [CrossRef] Crowley, T.J.; Choi, K.Y. Calculation of molecular weight distribution from molecular weight moments in free radical polymerization. Ind. Eng. Chem. Res. 1997, 36, 1419–1423. [CrossRef] Crowley, T.J.; Choi, K.Y. Optimal control of molecular weight distribution in a batch free radical polymerization process. Ind. Eng. Chem. Res. 1997, 36, 3676–3684. [CrossRef] Ross, R.T.; Laurence, R.L. Gel effect and free volume in the bulk polymerization of methyl methacrylate. Aiche. J. 1976, 72, 74–79.


33. 34. 35.

23 of 23

Fan, S.; Gretton-Watson, S.P.; Steinke, J.H.G.; Alpay, E. Polymerisation of methyl methacrylate in a pilot-scale tubular reactor: Modelling and experimental studies. Chem. Eng. Sci. 2003, 58, 2479–2490. [CrossRef] Li, R.J.; Henson, M.A.; Kurtz, M.J. Selection of model parameters for off-line parameter estimation. IEEE T Contr. Syst. T 2004, 12, 402–412. [CrossRef] Nowee, S.M.; Abbas, A.; Romagnoli, J.A. Optimization in seeded cooling crystallization: A parameter estimation and dynamic optimization study. Chem. Eng. Process. 2007, 46, 1096–1106. [CrossRef] © 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

Combining On-Line Characterization Tools with Modern Software - MDPI

Combining On-Line Characterization Tools with Modern Software - MDPI

Suggest Documents

Combining On-Line Characterization Tools with Modern Software ...

Combining On-Line Characterization Tools with Modern ... - MDPI

Combining Targeted Agents with Modern

CiCUTS: Combining System Execution Modeling Tools with ...

Combining Tools to Design and Develop Software Support for ...

Combining Pickering Emulsion Polymerization with Molecular ... - MDPI

An Introduction to Modern Software Development Tools Creating A ...

pdf-1295\modern-software-tools-for-scientific-computing-from-brand ...

pdf-1295\modern-software-tools-for-scientific-computing-from-brand ...

Combining Epidemiologic and Biostatistical Tools

Experiences with Commissioning Software Tools at ...

Combining Lean Concepts & Tools with the DMAIC Framework to ...

Combining Click-Stream Data with NLP Tools to Better Understand ...

Combining an ancient tradition with modern ... - Ingenta Connect

Combining Click-Stream Data with NLP Tools to Better Understand ...

Combining Click-Stream Data with NLP Tools to Better Understand

Combining Symbolic Tools with Interval Analysis. An Application to ...

Metallocyclodextrins: Combining Cavitands with ... - Wiley Online Library

Software Tools and Approaches for Compound Identification of ... - MDPI

Software Tools and Approaches for Compound Identification of ... - MDPI

Tools for modern developers - ALMtoolbox

MODERN SOFTWARE RENDERING

SOFTWARE TESTING TOOLS

Software Tools on Demand